**B6.**** Motion on precribed trajectory**

**b)****
Rigid body, which can rotate about a fixed axis, in motion**

**Duration of rotation. Angular
velocity. Measure of magnitude of angle. Trajectory velocity**

A body, which can rotate about
a fixed axis, remains at rest only when the resultant of the
forces acting on it intersects the axis. If *it does not**, *the body rotates.

At the beginning of a rotation,
*all* its points start to move *simultaneously* along circles, at the end, they stop *simultaneously*. Even if only one point of the body has
completed its circle once, the body regains the same position as
it had at the start. The time interval *T *between the start
and the end of a full rotation is called its *period*. The greater the distance of a point
from the axis, the longer its circular trajectory during *T*, whence it also has
a larger velocity. Since the body is rigid, the paths and
velocities of points at different distances from the axis are
related. The body's rigidity forces all points which, for
example, lie *duringrest *on a straight line, to do so also *during rotation**.*

*If *during rotation about the axis (Fig. 87) which is perpendicular to the plane of
the drawing at *c*, the
point *f* moves from *f*_{1} to *f* _{2}, and *e, d *and all other points
which lie on the* same *line as *f*, perpendicular to
the axis, simultaneously move to *e*_{2}*, d*_{2}*,
..., *then *e*_{2}*, d*_{2}*, ... *lie
on the* **same* line* *through* f*_{2}*
*and perpendicular to the axis.

In other words: for all points of the same line, the
angle described by their perpendicular distances from the axis *f*_{1}*
, e*_{1}*c, d*_{1}*c, ··· *during
this time is the same. However, because the body is rigid, the
same angle is also described by* *the perpendicular
distances from the axis of rotation by all other points, that is,
the *entire *body has turned during this time
interval by this angle. The velocity at which it does this, that
is, the ratio of the size of the angle over the time used is
called *angular velocity**. *We will return later on to this
topic.

The magnitude of
an *angle* (Fig. 87) measures the circular arc
between its legs in its ratio to the length of the corresponding *radius*: In this case, the ratio of *f*_{1}*f*_{2} to* f*_{1}*c *or*
c*_{1}*c*_{2 }to*
c*_{1}*c.* Since *f*_{1}*f*_{2}/*f*_{1}*c =
c*_{1}*c*_{2}/*c*_{1}*c = ···, **this
ratio is unique *for
that angle, whence it can be used as measure of the magnitude of
the angle. - The length of the arc, described by the radius of 1
cm about *c *is in the same ratio as *f*_{1}*f*_{2}* *to* f*_{1}*c, *etc. Denoting this arc length by *j*, then *f*_{1}*f*_{2/}*f*_{1}*c =
··· = **j* /1 = *j*, that is, the magnitude of the angle is measured by the
length of the circular arc, described with radius 1 cm about the
vertex of the angle. The magnitude of the angle of 360º is the
entire circumference of the circle with radius 1 cm. Hence

angle |
360º | 180º | 90º | 45º | ··· | |||||

arc |
2p | p | p/2 | p/4 | ··· |

The length of the circular arcs *f*_{1}*f*_{2},
*e*_{1}*e*_{2},
etc. related to this arc*j* are *f*_{1}*f*_{2} = *f*_{1}*c·**j, **e*_{1}*e*_{2}=*e*_{1}*c·**j*, etc.

The
path described by a
point *m*_{r}at distance *r* from the
axis of rotation is *r* times as long as the arc *j* , which
a point *m*_{1}_{
}describes at the
distance 1 cm from the axis during the same time interval, whence* **m*_{r}_{ }has
*r* times the velocity of *m*_{1}. The *velocity *of *m*_{1}_{ }is measured by the length
of the arc *j*, covered by it, in ratio to the time
used. However, this arc is also the measure of the angle w, by which the *body has* rotated during the same time interval.
The ratio of the length of the circular arc, covered by a point
at distance 1 cm from the axis, to the time used is therefore
simultaneously a measure for the velocity by which the *entire* body describes this angle. It is called
its *angular velocity**.*

*If a body has undergone a
complete turn*, it has
described the angle of 2p. If the duration of this turn is *T *sec, is has
described in 1 sec the angle 2p/*T*. However, it describes in 1 sec the angle *w* - according to the definition of its angular velocity -
whence *w* = 2p*/T. *If it undergoes in 1 sec *n* cycles, so
that *T = *1/*n, *then *w* = 2p*n*, where. *n* is the number of revolution
per sec.

The
*discussion of the
uniformity of velocity and acceleration*
explains the meaning of uniformity of angular velocity and
acceleration. The same considerations which apply to motion along
straight lines are also valid for circular arcs. If the point
describes always at 1 cm from the axis of rotation at uniform
angular velocity during unit time the arc of length w, then w is
the angular velocity of the body. You must distinguish between *angular velocity** *and *path*
velocity. The *path
velocity* is the ratio
of the length of the arc *r**j *covered during the time *t* used. The *angular velocity* of *any point*
is the same as for any other point, that is, for the entire body.
In contrast, the path velocity increases with *r*, that is,
the further away a point is from the axis, the greater its
velocity. If the *angular
velocity* of every point
is known, then also the *path
velocity of* every point
is known: It is equal to the product of *w* and its distance from the axis of
rotation.

We are especially interested in
points on *a sphere
rotating about a diameter*, because we live on such a surface - at least *we can conceive Earth to be such*. Earth rotates by 360º in one day = 86400 sec (that
is, half as fast as the hour-hand of a clock), its angular
velocity is therefore 360º/86400. that is, only ¼ angle
minute/sec - very small indeed (in arc measure: 2p/86400
= 0.000073). The* path *velocity of a point on its surface
depends on its geographical latitude *.* It is* **w* ·*r*cos*j**, *where at the equator *r =*
6 378 388 m; it is therefore at the equator 73·10^{-6}·6
378 388 465 m,
at latitude 50º 299m. Points at the same latitude have the same
path velocity; *it
changes with Latitude*.

You do not note directly that
the *path **velocities
**of* points at the surface of rotating
bodies, which lie at different distances from the axis of
rotation, differ [at least not with those of a rigid body, the *non-rigid *body deforms - a noticeable action of this distinction]. However, you
can note it *indirectly*, as a mobile mass - say a mass point -
moves on a rotating body from *one*
latitude to *another*. As a consequence of the rotation of
the body, the mass point follows then *relative to the points of the surface
*of the body a* *path which differs
from that which it would pursue if the body did not rotate. The
difference between the path velocities of points on the surface
at different latitudes manifests itself.

This *phenomenon is as well of interest* to us, because we live on the surface
of a rotating sphere and the water particles in rivers, the air
particles in the atmosphere as well as bullets of long distance
guns, etc. move from one circle of latitude to another; also the
rotation of the plane of oscillation of a pendulum is here of
concern as well as the deflection from the vertical of a freely
falling body. On Earth's surface, you note the deviations only
when the motion of mass points lasts a long time or when the
velocities are very large, because, as we will see, the product
of the velocity of the mass point and the angular velocity of the
rotating body contribute here, but the angular velocity of Earth
is very small.

However, on the surface of a
body which we can rotate sufficiently fast, the deflections
become noticeable quickly enough. - Following its discoverer Coriolis, the *deflection
motion *- an
acceleration - is called *Coriolis** acceleration* and the force, by the action of which
it is sensed, *Coriolis** Force*.

A readily understood *special case *demonstrates the generation of the
deviation (Fig. 88), A disk rotates in the direction of the arrow
in the plane of the figure about the vertical through its centre *M*
and on it moves a mass point starting from *M*. This point
moves frictionless over the disk, that is, it follows only its
inertia. Therefore we imagine it to move *just above* the disk parallel to it along a *radial straight line* at rest like on a rail. It moves at a
constant velocity *c *starting at* M*.

At the end of the time interval *t*_{1}, it reaches that point of the rail
underneath which itwas located *initially*
at the point *B *of the disk; however, *during t*_{1}, the disk has turned away by the angle *a *below the material point so that the
point of the disk, which initially was at *B*, is at *B*_{1}*.*

At the end of the next time interval* t*_{2}*
= t*_{1}, the
material point reaches on the rail that point, underneath which
the point *C* of the disk was located initially; during the
time interval *t*_{2 }the disk has turned by the
same angle *a*, whence the point of the disk,
initially at *C*, is at *C*_{2} , etc. (The
rotating disk and the rail at rest behave with respect to each
other like the dial-plate and the hand of a horizontally placed
watch, except that the dial-plate rotates and the hand rests).

The *sidewards deviation *of a point of the disk from the line,
along which moves the material point, for example, the deflection
*CC*_{2} of the disk point *C *is given by the
product of the angle *CMC*_{2}, measured as an arc,
and the segment *MC*. The angle *CMC*_{2} is *w **t *and the segment has the length *ct,*
the deviation *CC*_{2}* *is therefore *s = c**w** t *= (2*c**w */2)*t*^{2}. Obviously, this
is a distance which has been covered with the acceleration 2*c**w* during the time *t *(just imagine
that you have placed next to it a distance *fallen* at the acceleration *g*!). The
points of the disk deviate in the direction of the rotation of
the disk with a velocity which is perpendicular to the direction
of the velocity *c* of the material point and has the
acceleration 2*c**w*.

Discussion of Fig. 89: About the Coriolis deviation. (Coriolis) In the plane* **of
the drawing* rotates with constant angular velocity *w *a horizontally oriented disk about the
vertical through *M. *From *M *a missile is launched
radially at the target *x* resting on the disk with constant
velocity *c*. The *rotation**
*of the disk deflects the points of the *disk*,
over which the missile passes and the *target*
towards the left side, so that the missile hits the disk to the *right of* the target. An observer
who rests on the disk, that is an observer who partakes in the
motion of the disk and does *not* sense it, draws the
conclusion: "*A force has diverted
the missile towards the right hand side.*" This *apparently present deflecting force is*
the Coriolis force
. - The solid curve is the trajectory of the missile relative to
the disk, which is assumed to be *at rest*,
that is, for an observer, travelling with the disk, the *apparent *path of the missile; the
broken curve, displays the a*ctually
displaced points of the disk* due to the rotation
during the motion of the missile.

This is how an observer on a immobile line
imagines the movement of the material point. It appears quite
differently when seen from the disk by an observer resting on it,
that is, who takes part in the motion of the disk, but *does not sense it*! (Note the last
point; just imagine: We take part in the
rotation of Earth without noting it!) In order to get insight
into what* **this** *observer sees, imagine a missile, just above a
plane and flying parallel to it from the centre along a straight
trajectory, the plane is infinite in all directions, nowhere are
there points of reference, relative to which you can sense the
rotation of the plane. The process will then appear to an
observer who rests on this plane as discussed with Fig. 89 above.
We ourselves are in just
that position (at the centre of the
horizontal plane, which rotates about us.)

We are
mainly interested in the *Coriolis** deviation *because we live on the surface of rotating Earth and can
explain by means of it certain *changes in direction* of
geophysical processes on its surface. A mass which moves along
the surface of Earth (air in winds, water in rivers) does not
move without friction and therefore the centrifugal force also
contributes. But these two phenomena are so little active that we
may neglect friction as we have initially assumed. But we must note one point: At
a point of Earth's surface at latitude *j*, we must only take into account the angular
velocity's component *w *·sin*j* .Among the
geophysical processes in which the Coriolis Deviation has a role is
first of all the *deflection
of wind as a result of Earth's rotation*.

The steady difference between the heat transfer
at higher and lower latitudes generates a *basic circulation in the atmosphere**. *The excess heat received by the equatorial
region induces a circulation: In the upper layers of the
atmosphere, the winds move *polewards*
from the Equator, they move simultaneously
in the *lower* layers from the Poles to the Equator. But *Earth's rotation complicates the
process*. The upper, polewards moving
air approaches gradually the axis of rotation, increases thereby
in the upper latitudes its velocity *relative to* points on
Earth's surface, eventually moves *ahead of them* and becomes a
West wind. For a similar reason becomes the lower, to the Equator
flowing air (because it remains behind the points on Earth's
surface) in higher latitudes an East wind. That was in 1930
(since Ferrel 1860) the basic concept of the cause of the *deflection of wind due to Earth's
rotation*.

Baer has explained peculiarities
of the formation of the shores of meridi0nally running rivers in
Russia by the *Coriolis Deviation (**Baer**'s
Law)*: He found that in Siberia the right shores of
many rivers in lowlands were for long distances *steep and high*, the left banks *low and flat*. He explained this
phenomenon by the increased pressure of the current towards the
shore due to Earth's rotation; it forces the river to shift its
shore towards the right side as long as no higher land opposes it

The flowing water, which moves
from the South to North brings with it higher velocities than
prevail in Northern regions of Earth and thus presses against the
Eastern shores, because *Earth rotates
towards the East*, and with it the excess water
which comes from the lower to the higher latitudes. Conversely,
water flowing from the polar regions towards the Equator will
arrive there at lower speeds and press against the Western banks.
However, on the Northern hemisphere, the Eastern shore of rivers
flowing North is on the right and those of river flowing South on
the left. Hence on the Northern hemisphere are the Eastern shores
of rivers, which flow more or less meridionally, the
affected, *steeper and higher ones*,
the Western shores *low and flooded*,
and indeed more so as the directions of rivers approach
Meridians, so that along rivers and river sections which are
almost totally meridional other disturbing effects show off less.
- *If this explanation is correct*,
the situation on the Southern must be reverse. Von Baer has shown that this is the case.

Also rotation is visualized ** **by a directed segment. It is placed along the axis of
rotation, its length equals the angular velocity *w *and it is
given an arrow in the direction which forms with the sense of
turn a *right- hand screw** *(Fig. 90).

The *rotor* is an axial vector (in contrast to a polar vector). It
is unimportant, from which point* O *of the axis the vector
is drawn. - Moreover, the connection of the path velocity of a
point with the angular velocity of a body can be envisaged. A
point *P *which is* *at distance *r*_{1}
from the axis and at the instant is on the ray *r* from *O
*to *P* and makes the angle *a *with the
vector *w*, has the path velocity *v *= *w**r*_{1}*=** w**r*sin*a*. On the
right hand side, you have twice the area of the triangle, formed
by *w *and *r.* It is called the *vector product* and represented by the directed segment *v*.

You make the magnitude of *v* equal to
twice the area of the triangle and place it perpendicular to the
plane of the triangle in such a manner that the arrow and *a* form a
right hand screw , which moves the vector *w* along the
*shortest path* into the direction of the vector *r*.

At times, a body may *rotate simultaneously about several
axes*. This vector representation
allows its rotation about several axes to be composed into
one about a *resultant axis**.* What is
the meaning of simultaneous rotation about several axes?

You should think here of the top with which you played as a child,
the tip of which remains in place while it rotates (Fig. 91): It rotates **1.** about its *axis of symmetry* (axis of figure) and simultaneously **2. **this axis
rotates about the *vertical
axis*, passing through its point of
support, and describes a *cone
*(the point of the top is the tip, the
vertical the axis of the cone.) As the top slows down, its axis
swings in addition towards the vertical and back, that is, **3. **it
turns about a *horizontal
line through its support point*. In other words:
Every mass point of the body - only not the point of support - *rotates simultaneously about several
axes*. All these rotations combine into
the motion of the *staggering
top* which struggles with falling over.
The axis of symmetry of the top is no longer the preferred axis
of rotation of the system, it is only *one** *of
several*.*

How to combine rotations? How
to find the resultant rotation and its axis?
We will discuss here only the *fundamental rule*
according to which two rotations about two axes which pass
through the same point are combined into a resultant rotation
(and how a rotation is decomposed into two components). Let *u*_{1}
and *u*_{2} be the two vectors of two rotations. Each determines the axis and angular
velocity of corresponding rotations: Each
of these rotations gives every mass point of the body a definite
velocity. The composition of the *rotations** *tells you
the *velocities** *which the mass points of the body have,
generated by these two rotations. **1. **Also the *resultant motion *is a rotation of the body about an axis; **2.** The *angular velocity and direction of
the axis *of the resultant rotation is
given by the *rotor* which is the result of addition of the vectors *u*_{1}
and *u*_{2 }according to the parallelogram law.

*By the same law*, you
can decompose a rotation about an axis into several simultaneous
rotations (about several axes). An example follows: It may prove
useful to conceive the rotation of Earth as having beern
decomposed into two components, in fact, whenever you want to
compute processes, the cause of which is the rotation of Earth's
axis and the magnitude of which depends on the Latitude of the
site of observation. Examples are *Foucault**'s
pendulum experiment*, *Hagen**'s experiment *to prove the rotation of Earth by the isotomeograph,
etc.

You represent Earth's angular
velocity *w
*(Fig. 92) by the
vector ** v **which lies along the
axis of Earth and points northwards from Earth's centre; its
length corresponds to the angle 2p/86400, covered during one
second. ( In the numerator is the angle covered during a mean Sun
Day, measured on a circle with radius 1, in the denominator the
number of seconds per day). You now

For example, this decomposition of Earth's rotation shows why Foucault's pendulum and the Isotomeograph demonstrate best at the poles the turning of Earth's axis, and not at all at the Equator.

**Centripetal and centrifugal force**

If the body turns due to the action of a force,
say, like the hand of a clock, as the force stops to act, it
continues to turn in the same direction and, indeed, with that
angular velocity which it had as the force stops.Therefore *rotation with a uniform angular
velocity *is an inertial motion: The
body maintains its state of motion in direction and angular
velocity, unless an external force stops it (a possibility which
will be excluded here). But this *quasi-inertial motion differs *from the *real* *inertial
motion*: The individual mass points
move with uniform velocity, but in *circles**. *However, a
point can *leave** *a straight line only if a force acts. Hence we
must *assume** *that a force acts and indeed deflects it
radially towards the centre of the circle which it describes from
that straight line which it *would
have* followed had it been able to act
inertially - this is the *centripetal
force**.*

The force resists inertially
the action of this deflecting force, diametrically opposite to the deflecting force: This resistance attempts to inhibit an approach of the point
towards the centre of the circle; it acts radially, but away from
the centre, whence it is called the *centrifugal
force*. This centrifugal force has
therefore only *significance
*relative to the simultaneously acting
centripetal force. It is the *reaction*, if you consider
the centripetal force to be the *action*; as always in the
case of action and reaction, it occurs concurrently with the
centripetal force, that is, it must appear and vanish
simultaneously with it and also for the same reason be equally
large.

In order to *understand how the centripetal force acts*, you must understand the relationship of a mass point
to the centre of its trajectory. Fig. 93 shows a plane at right
angle to the axis of the body; *C *is its intersection with
the axis, the circle is the path of a point and the curved arrow
the direction of the rotation. The *mass point*, due to its
inertia, wants to continue along the tangent, the *centre*,
as a point of the fixed axis, must stay in place. Both belong to
the same rigid body, whence they cannot change the distance
between them. The centre is forced to *stay in place* and not change its distance to the mass point, the mass
point not to change its distance from* C, *but nevertheless
to advance. The circulating mass point therefore tends to
displace the centre (maintaining its distance to it).

Since you can imagine its point of attack
shifted anywhere along its direction, the centrifugal force also
tends to increase the distance *between *the mass points and
the centre. It thus competes with the forces of *rigidity**
*of the body and its *cohesion*. If the body were perfectly rigid, this property would make the centrifugal force ineffective, however, no body
is totally rigid. The mass points move therefore more or less
under the action of a *sufficiently
large** *centrifugal force, that
is, the body gives in, deforms and can even split up. You can
envisage those forces which are essential for the rotation of a
rigid body (centripetal and centrifugal forces, action on the
centre, competition between cohesive forces) by a meaningful
process, which however disregards rigidity, but realizes
the *essential** **aspects
of rotation**. *

If you
swing by hand a heavy body on a rope in a circle,
the rope becomes stretched as a result of the centrifugal force
and the hand senses a pull towards the body. It must pull it
strongly inwards (*centripetrally*), in order not to be oulled outwards (*centrifugally*). If you swing the body faster and faster, you increase
the centrifugal force. The pull, felt by the hand, becomes
larger, the rope tighter and eventually it breaks, the motion in
a circle and the pull on your hand end *simultaneously** *and the body flies off.